The pickaxe is inside the prison, behind a door at the bottom of the stairs next to the ones you came in by. Get into the prison first, bring an iron rod for that door, and keep an eye on the skeleton archers above you.
How do you reach the prison for the pickaxe?
Bring the Small Key and Prison Key, then use the wooden plank to cross from the hub toward the prison entrance. If you're still sorting out the locked doors, follow the Prison Key route before heading over.
- Go up the stairs in the hub, turn left, then left again to reach a large wooden plank.
- Drag the plank into place as a bridge and cross to the other side.
- Open the first door with the Small Key, then go down the spiral staircase.
- Burn away the webbing at the bottom with a torch equipped in your right hand. An offhand torch won't do this job.
- Unlock the second door with the Prison Key to enter the prison.
Where do you find the pickaxe in the prison?
Return to the stairs you first used to enter the prison, then take the stairs beside them farther down. Open the door below to reach the pickaxe.
You'll need an iron rod for that door. If you're out, see how to get iron rods. Grab the pickaxe and move: the skeleton archers above have a clear shot while you stand there.

What can you break with the pickaxe?
Yellow goo. That includes the goo blocking passages in the prison and the Goop piles in the sewers, which drop clay.
If you hit Rock pile one and see 0 damage, stop swinging. The pickaxe doesn't break that rock pile, so leave it and come back later.